An Assessment of the Accuracy of Multireference Configuration

assessed for bond-breaking reactions in BH, HF, and CH4 by comparison to the full configuration interaction limit. These results allow one, for the fi...
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J. Phys. Chem. A 2003, 107, 5611-5616

5611

An Assessment of the Accuracy of Multireference Configuration Interaction (MRCI) and Complete-Active-Space Second-Order Perturbation Theory (CASPT2) for Breaking Bonds to Hydrogen Micah L. Abrams and C. David Sherrill* Center for Computational Molecular Science and Technology, School of Chemistry and Biochemistry, Georgia Institute of Technology, Atlanta, Georgia 30332-0400 ReceiVed: March 15, 2003; In Final Form: May 8, 2003

Complete-active-space self-consistent field (CASSCF), complete-active-space second-order perturbation theory (CASPT2), and two restricted active-space variants of multireference configuration interaction (singles, doubles, and limited triples and quadruples, CISD[TQ], and second-order configuration interaction, SOCI) have been assessed for bond-breaking reactions in BH, HF, and CH4 by comparison to the full configuration interaction limit. These results allow one, for the first time, to ascertain typical errors for such reactions across the entire potential energy curve. They also provide an assessment of different prescriptions for choosing an active space. A valence active space and a one-to-one active space were considered along with the basis sets ccpVQZ, 6-31G**, and 6-31G* for BH, HF, and CH4, respectively. The valence active space performs better than the one-to-one active space for BH but is inferior for HF. Always choosing the larger of the two active spaces for a given molecule leads to the best results. When using the larger of the two active spaces, the nonparallelity errors for CASPT2, CISD[TQ], and SOCI were less than 3.3, 1.4, and 0.3 kcal mol-1, respectively. These results are superior to those of unrestricted coupled-cluster with perturbative triples [UCCSD(T)] for these same systems.

1. Introduction The last decade has seen a major advancement in electron correlation methods based on a multiconfiguration wave function, that is, multireference versions of configuration interaction (MRCI), perturbation theory (MRPT), and coupled-cluster theory (MRCC).1 Although the idea of such methods is rather old,2-4 significant new approximations and algorithms1,5-14 have been developed. While these methods remain too computationally expensive to use on molecules with more than a few heavy atoms, they are nonetheless the only methods capable of accurately describing many chemical processes, particularly bond-breaking and bond-forming reactions. Multireference configuration interaction has been the standard model for determining accurate potential energy surfaces of polyatomic molecules for the last 30 years. Problems with size consistency still remain, but thus far, alternative size-consistent methods are not yet in common use for generating potential energy surfaces of spectroscopic accuracy. Multireference perturbation theory has been applied to a number of chemical problems including molecular structure, electronic spectra, and transition metal chemistry. Perhaps the most popular variation of MRPT is the complete-active-space second-order perturbation theory (CASPT2) method of Andersson and Roos.5 Errors in geometries, binding energies, and excitation energies have been systematically studied;15 however, the error in CASPT2 along a full potential energy curve has not been thoroughly evaluated. The most straightforward way to determine the error of a given correlation model is to compare it to the exact solution of the electronic Schro¨dinger equation for the given one-electron basis, which is the full configuration interaction (FCI) result. A series of studies by Bauschlicher in the late 1980s provided FCI energies at a few geometries along the potential energy

curves of several small molecules.16-20 Advances in CI algorithms and computer hardware have made it possible to obtain more complete FCI potential energy curves21-26 for some simple systems. For example, Olsen and co-workers studied polarized double-ζ FCI potential energy curves for bond breaking in several electronic states of the N2 molecule24,27 and the symmetric dissociation (breaking both bonds) of H2O. The benchmark FCI results were compared to perturbation theory and coupled-cluster models to indicate how these approximate methods perform for very challenging cases. Such benchmarks are essential for the calibration of new theoretical models meant to describe bond-breaking processes.28-33 In the present study, we compare to FCI potential energy curves for three molecules (BH, HF, CH4) in which a bond to hydrogen is broken. This should represent a common, chemically important process that one might expect to be the easiest type of bond-breaking reaction for standard quantum chemical methods to describe accurately. However, we have recently shown26 that single-reference methods, even when based on an unrestricted Hartree-Fock reference, are not very accurate for these simple systems; unrestricted coupled-cluster with perturbative triple excitations [UCCSD(T)] yields nonparallelity erorrs of about 4 kcal mol-1. It is therefore of interest to compare the performance of multireference approaches for these molecules. Here, we assess the popular complete-active-space selfconsistent field (CASSCF)34,35 method and complete-activespace second-order perturbation theory (CASPT2) with the nondiagonal zero-order operator.5 We also consider two multireference configuration interaction singles and doubles (MRCISD) methods. One of these, the second-order CI (SOCI),2 generates all possible singly and doubly substituted configurations from every active space configuration that can be formed

10.1021/jp034669e CCC: $25.00 © 2003 American Chemical Society Published on Web 06/26/2003

5612 J. Phys. Chem. A, Vol. 107, No. 29, 2003 by distributing the active electrons among the active orbitals. This is perhaps the most complete type of a MR-CISD wave function. The second MR-CISD approach considered here is the CISD[TQ] wave function of Schaefer and co-workers,36,37 which generates all single and double substitutions from the reference set of all singly and doubly substituted configurations that can be formed in the active space. As such, CISD[TQ] may be thought of as an approximation to SOCI in which all configurations that are more than quadruply substituted (relative to the Hartree-Fock reference) are discarded. Previous high-quality benchmarks for the molecules considered in this study include a cc-pVTZ FCI potential energy curve for BH,38 a cc-pVDZ FCI potential energy curve for HF,38 and 6-311++G(df,p) multireference CI results39 for the breaking of a single C-H bond in CH4. We have previously assessed26 several single-reference correlated methods by comparison to FCI for BH, HF, and CH4 in the aug-cc-pVQZ, 6-31G**, and 6-31G* basis sets, respectively, and we will refer to the FCI results of that study in the present work. Because of additional difficulties in converging CASSCF wave functions with basis sets containing many diffuse functions, for BH we will compare to cc-pVQZ FCI results, which were also generated in our previous study. 2. Theoretical Approach Here, we use the cc-pVQZ, 6-31G**, and 6-31G* basis sets for BH, HF, and CH4, respectively.40 We have examined CASSCF,35,34 CISD[TQ],37 SOCI,2 and CASPT2 with the nondiagonal zero-order operator.5 The CISD[TQ] and SOCI computations were performed using CASSCF orbitals. Two orbital active spaces were used in this study. The first active space is a valence active space, by which we mean that there is one active space orbital of a given irreducible representation for each molecular orbital of the same irreducible representation that can be formed from the valence orbitals on the atoms in the molecule. The valence orbitals of the atoms are of course the usual 1s for hydrogen and 2s, 2px, 2py, and 2pz for B-F. The valence active space is commonly used in CASSCF computations, where it is sometimes referred to as full-valence CASSCF. This approach was also introduced by Ruedenberg under the name full optimized reaction space (FORS).35 The second active space, which we will refer to as one-toone or 1:1, includes the occupied valence orbitals plus an active virtual orbital (of the same symmetry) for every occupied valence orbital. This is the active space used in generalized valence bond perfect pairing computations, and it is being considered in recent work by Head-Gordon and co-workers.41,42 Its use in CASSCF computations dates back at least to a 1980 paper by Roos, which shows much better agreement with experiment than full-valence CASSCF for equilibrium properties of the water molecule.43 The benefit of these two choices of active spaces, full-valence and one-to-one, is that they are a priori definitions that can be used to define the active space of a molecule in the absence of any preliminary computations or arbitrary choices of thresholds. These two active spaces were previously compared for the more challenging case of double dissociation in H2O by Olsen and co-workers.22 More specifically, the active spaces for the molecules in the study are as follows: BH val ) (4e-/3011) and 1:1 ) (4e-/ 4000); HF val ) (8e-/3011) and 1:1)(8e-/4022); and CH4 val ) 1:1 ) (8e-/62), where the notation indicates (number of active electrons/number of active orbitals per irrep of the largest Abelian subgroup). For CH4, definitions of the valence and 1:1

Abrams and Sherrill

Figure 1. Potential energy curves (hartree) for BH using the cc-pVQZ basis set.

active spaces are equivalent. Note that the valence active space is larger than the 1:1 active space for the BH molecule, but it is smaller for HF. The core 1s orbitals were constrained to remain doubly occupied in all cases. The CASPT2 calculations were performed with MOLCAS 5.2.44 All other calculations were performed with the DETCAS and DETCI11 modules of PSI 3.2.45 The potential energy curve for methane was obtained by constraining three C-H bonds to their equilibrium bond length (1.086 Å)46 and the HCH angles at the tetrahedral value while stretching a single C-H bond. Full CI results are taken from our previous work.26 3. Results and Discussion The potential energy curves for BH are shown in Figure 1. From the figure, it is clear that all of the methods considered here provide qualitatively correct potential energy curves. This contrasts to the behavior of many single-reference correlation methods, evaluated previously.26 Although the CASSCF potential energy curves are significantly higher in energy than the exact FCI curve (because of the limited treatment of electron correlation), they have approximately the correct shape. The CASPT2 curve with the smaller active space is also significantly higher than the FCI curve, but again it has the correct shape. The other methods considered are accurate enough that they are hard to distinguish from the FCI curve at this scale. Because the qualitative behavior of the potential energy curves is similar for the other two molecules, we omit figures of the curves for HF and CH4. More instructive are plots of the errors versus FCI as a function of bond length, which are presented in Figures 2, 3, and 4 for BH, HF, and CH4, respectively. These are discussed below. Tables 1, 2, and 3 contain the error versus FCI, and Tables 4, 5, and 6 contain the maximum, minimum, and nonparallelity errors (NPE) for BH, HF, and CH4, respectively. The NPEs are computed as the difference between the maximum and minimum errors along the potential energy curve, and they provide a measure of how well each method mimics the overall shape of the exact FCI potential energy curve. A NPE of zero would imply a complete match to the shape of the FCI curve. It is well-known that the CASSCF wave function provides qualitatively correct potential energy surfaces if a proper active space is chosen, because it describes the interaction between important near-degenerate configurations, called nondynamical correlation. However, the method does not contain enough configurations to accurately describe the usual short-range electron-electron repulsions, called dynamical correlation.

Assessment of the Accuracy of MRCI and CASPT2

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TABLE 1: FCI Energy and Error Versus FCI (hartree) for BH Using the cc-pVQZ Basis Set R(B-H), Å

FCI

CASSCF (val)

CASSCF (1:1)

SOCI (val)

SOCI (1:1)

CASPT2 (val)

CASPT2 (1:1)

0.80 0.90 1.00 1.10 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00 4.20 4.40 4.60 4.80

-25.081 355 -25.162 408 -25.206 390 -25.227 709 -25.235 155 -25.228 283 -25.209 491 -25.188 268 -25.168 318 -25.151 002 -25.136 762 -25.125 628 -25.117 376 -25.111 577 -25.107 695 -25.105 194 -25.103 629 -25.102 666 -25.102 081 -25.101 726 -25.101 512 -25.101 383 -25.101 305

0.052 543 0.052 480 0.051 187 0.049 855 0.048 658 0.046 621 0.044 943 0.043 511 0.042 268 0.041 200 0.040 307 0.039 581 0.039 010 0.038 575 0.038 256 0.038 032 0.037 878 0.037 775 0.037 706 0.037 660 0.037 629 0.037 609 0.037 594

0.081 702 0.081 161 0.080 188 0.079 181 0.078 273 0.076 734 0.075 332 0.073 798 0.071 984 0.070 005 0.068 116 0.066 450 0.065 052 0.063 942 0.063 116 0.062 538 0.062 151 0.061 901 0.061 740 0.061 638 0.061 573 0.061 531 0.061 503

0.000 543 0.000 536 0.000 496 0.000 490 0.000 479 0.000 447 0.000 408 0.000 365 0.000 325 0.000 289 0.000 256 0.000 224 0.000 193 0.000 164 0.000 138 0.000 119 0.000 106 0.000 097 0.000 091 0.000 088 0.000 087 0.000 086 0.000 086

0.003 673 0.003 764 0.003 628 0.003 451 0.003 286 0.002 988 0.002 717 0.002 477 0.002 256 0.002 032 0.001 833 0.001 676 0.001 557 0.001 471 0.001 410 0.001 369 0.001 342 0.001 325 0.001 315 0.001 308 0.001 304 0.001 302 0.001 300

0.008 719 0.008 986 0.008 822 0.008 685 0.008 565 0.008 338 0.008 073 0.007 742 0.007 346 0.006 895 0.006 396 0.005 871 0.005 366 0.004 929 0.004 589 0.004 343 0.004 174 0.004 065 0.004 002 0.003 975 0.003 962 0.003 956 0.003 957

0.022 021 0.022 119 0.021 979 0.021 784 0.021 599 0.021 225 0.020 788 0.020 088 0.019 317 0.018 594 0.017 990 0.017 526 0.017 217 0.017 045 0.016 965 0.016 933 0.016 923 0.016 921 0.016 921 0.016 922 0.016 922 0.016 922 0.016 922

Figure 2. Error versus FCI (kcal mol-1) for BH using the cc-pVQZ basis set. The cc-pVQZ FCI equilibrium bond distance is 1.2260 Å.

Figure 3. Error versus FCI (kcal mol-1) for HF using the 6-31G** basis set. The 6-31G** FCI equilibrium bond distance is 0.9214 Å.

Because more electrons are closer together near equilibrium than at the dissociation limit, the degree of dynamical correlation is larger there and so are the CASSCF errors. This is seen quantitatively in Tables 1-3. The only exception is for HF with the 1:1 active space, for which the CASSCF error is fairly constant and rises slightly with distance. Here, the 1:1 active

Figure 4. Error versus FCI (kcal mol-1) for CH4 using the 6-31G* basis set. The equilibrium bond distance is 1.086 Å from ref 46.

space is sufficient to capture a significant portion of the dynamical correlation near equilibrium. Given that the degree of dynamical correlation is larger near equilibrium, one would expect that increasing the size of the active space would cause a greater improvement in the error at equilibrium than at dissociation. The nonparallelity errors in Tables 4 and 5 verify this expectation, keeping in mind that the valence active space is larger for BH but smaller for HF. Enlarging the active space from 1:1 to valence for BH decreases the error at 1.2 Å by 18.6 kcal mol-1 and at 4.8 Å by 15.0 kcal mol-1. Likewise, for HF enlarging the active space from valence to 1:1 reduces the error at 0.9 Å by 63.5 kcal mol-1 and at 4.0 Å by 41.2 kcal mol-1. Across all three molecules, the CASSCF nonparallelity error is as small as 4.8 kcal mol-1 and as large as 18.0 kcal mol-1. This is roughly comparable to the NPEs for CCSD based on RHF orbitals (8-13 kcal mol-1) and represents a definite improvement over UHF or UMP2 but not UCCSD, UCCSD(T), or UB3LYP.26 However, for more challenging bondbreaking cases, CASSCF should continue to work as well, while the quality of the single-reference approaches will degrade, as demonstrated by Olsen et al. for the simultaneous breaking of both bonds in H2O.22 We plan to examine bond breaking in additional molecules in future work.

5614 J. Phys. Chem. A, Vol. 107, No. 29, 2003

Abrams and Sherrill

TABLE 2: Error Versus FCI (hartree) for HF Using the 6-31G** Basis Set R(H-F), Å

CASSCF (val)

CASSCF (1:1)

CISD[TQ] (1:1)

SOCI (val)

SOCI (1:1)

CASPT2 (val)

CASPT2 (1:1)

0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00

0.162 876 0.163 972 0.164 811 0.165 401 0.165 754 0.165 890 0.165 824 0.163 754 0.159 260 0.153 437 0.147 734 0.143 307 0.140 478 0.138 900 0.138 081 0.137 669 0.137 462 0.137 360 0.137 309 0.137 285 0.137 273 0.137 268

0.063 955 0.064 196 0.064 369 0.064 491 0.064 589 0.064 691 0.064 822 0.065 908 0.067 861 0.069 764 0.070 855 0.071 317 0.071 503 0.071 590 0.071 633 0.071 651 0.071 655 0.071 654 0.071 651 0.071 648 0.071 646 0.071 645

0.000 867 0.000 886 0.000 904 0.000 920 0.000 936 0.000 951 0.000 967 0.001 045 0.001 147 0.001 267 0.001 421 0.001 593 0.001 756 0.001 891 0.001 993 0.002 069 0.002 125 0.002 167 0.002 197 0.002 218 0.002 232 0.002 242

0.009 195 0.008 197 0.007 510 0.007 037 0.006 706 0.006 469 0.006 293 0.005 878 0.005 585 0.005 257 0.004 894 0.004 572 0.004 347 0.004 216 0.004 146 0.004 112 0.004 095 0.004 089 0.004 088 0.004 090 0.004 094 0.004 133

0.000 716 0.000 726 0.000 733 0.000 738 0.000 742 0.000 744 0.000 746 0.000 756 0.000 761 0.000 747 0.000 731 0.000 719 0.000 712 0.000 709 0.000 707 0.000 706 0.000 706 0.000 706 0.000 705 0.000 705 0.000 705 0.000 705

0.011 912 0.010 698 0.009 819 0.009 168 0.008 688 0.008 328 0.008 056 0.007 540 0.007 543 0.007 666 0.007 659 0.007 561 0.007 494 0.007 484 0.007 501 0.007 521 0.007 537 0.007 551 0.007 565 0.007 556 0.007 576 0.007 579

0.005 343 0.005 374 0.005 404 0.005 434 0.005 466 0.005 504 0.005 552 0.005 872 0.006 140 0.005 996 0.005 766 0.005 612 0.005 529 0.005 494 0.005 483 0.005 481 0.005 484 0.005 485 0.005 486 0.005 486 0.005 486 0.005 486

TABLE 3: Error Versus FCI (hartree) for CH4 Using the 6-31G* Basis Set R(C-H), Å

CASSCF (val ) 1:1)

CISD[TQ] (val ) 1:1)

SOCI (val ) 1:1)

CASPT2 (val ) 1:1)

0.70 0.80 0.90 1.00 1.10 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00

0.082 722 0.081 922 0.080 894 0.079 746 0.078 547 0.077 354 0.075 191 0.073 614 0.072 781 0.072 627 0.072 942 0.073 469 0.073 997 0.074 414 0.074 697 0.074 872 0.074 971 0.075 024 0.075 051 0.075 064

0.001 716 0.001 733 0.001 722 0.001 696 0.001 663 0.001 628 0.001 583 0.001 601 0.001 702 0.001 885 0.002 137 0.002 429 0.002 724 0.002 991 0.003 213 0.003 384 0.003 510 0.003 599 0.003 659 0.003 699

0.001 550 0.001 551 0.001 523 0.001 475 0.001 415 0.001 350 0.001 226 0.001 134 0.001 082 0.001 061 0.001 060 0.001 069 0.001 081 0.001 091 0.001 099 0.001 104 0.001 107 0.001 109 0.001 110 0.001 110

0.012 026 0.012 138 0.012 128 0.012 034 0.011 883 0.011 698 0.011 309 0.010 986 0.010 742 0.010 533 0.010 326 0.010 122 0.009 943 0.009 815 0.009 734 0.009 689 0.009 667 0.009 657 0.009 652 0.009 650

TABLE 4: Maximum, Minimum, and Nonparallelity Error (kcal mol-1) for BH Using the cc-pVQZ Basis Seta method

max error

min error

NPE

CASSCF (val) CASSCF (1:1) SOCI (val) SOCI (1:1) CASPT2 (val) CASPT2 (1:1)

32.97 (0.80) 51.27 (0.80) 0.34 (0.80) 2.36 (0.90) 5.64 (0.90) 13.88 (0.90)

23.59 (4.80) 38.59 (4.80) 0.05 (4.40) 0.82 (4.00) 2.48 (4.60) 10.62 (3.60)

9.38 12.68 0.29 1.54 3.16 3.26

a

Values in parentheses indicate the corresponding bond distance (Å).

CISD[TQ] is a restricted active space (RAS)47 variant of MRCI in which RAS I is composed of active occupied orbitals and RAS II is composed of the active unoccupied orbitals. The inactive virtual orbitals comprise RAS III. The configurations are selected according to the following criteria: a maximum of two electrons are allowed in RAS III, and a maximum of four holes are allowed in RAS I. There are no restrictions on the occupancy of the orbitals in RAS II. These rules are equivalent to generating a MR-CISD in which the reference configurations are all singles and doubles within the active space. Results are not given for CISD[TQ] for BH because for four valence electrons, CISD[TQ] is equivalent to SOCI (discussed below).

TABLE 5: Maximum, Minimum, and Nonparallelity Error (kcal mol-1) for HF Using the 6-31G** Basis Seta method

max error

min error

NPE

CASSCF (val) CASSCF (1:1) SOCI (val) SOCI (1:1) CISD[TQ] (1:1) CASPT2 (val) CASPT2 (1:1)

104.10 (0.95) 44.96 (2.80) 5.77 (0.70) 0.48 (1.40) 1.41 (4.00) 7.47 (0.70) 3.85 (1.40)

86.14 (3.80) 40.13 (0.70) 2.57 (3.00) 0.44 (2.40) 0.54 (0.70) 4.70 (2.20) 3.35 (0.70)

17.96 4.83 3.20 0.04 0.87 2.77 0.50

a

Values in parentheses indicate the corresponding bond distance (Å).

TABLE 6: Maximum, Minimum, and Nonparallelity Error (kcal mol-1) for CH4 Using the 6-31G* Basis Seta method

max error

min error

NPE

CASSCF (val ) 1:1) CISD[TQ] (val ) 1:1) SOCI (val ) 1:1) CASPT2 (val ) 1:1)

51.91 (0.7) 2.32 (4.0) 0.97 (0.7) 7.62 (0.8)

45.57 (2.0) 0.99 (1.4) 0.67 (2.0) 6.06 (3.6)

6.34 1.33 0.30 1.56

a

Values in parentheses indicate the corresponding bond distance (Å).

Likewise, for the small valence active space in HF (only one active virtual orbital), again CISD[TQ] and SOCI are equivalent. It has been shown37 that the error versus FCI for CISD[TQ] along a potential energy curve is the opposite of CASSCF: the maximum error occurs near dissociation, while the minimum error occurs near equilibrium. Figures 3 and 4 verify this trend. The error at 0.7 Å is quite small for HF and CH4 (0.5 and 1.1 kcal mol-1, respectively). The error begins to increase at 1.0 Å for HF but decreases slightly for CH4 before beginning to climb at 1.8 Å. The nonparallelity error does achieve chemical accuracy (1.0 kcal mol-1) for HF. In both cases, the NPE is 2-4 times less than that obtained with UCCSD(T).26 The second-order CI wave function can also be defined in the restricted active space framework by placing the active space orbitals in RAS II and inactive virtual orbitals in RAS III. It is not necessary to employ the RAS I orbital space in this case. Configurations are selected by allowing all possible distributions of electrons in RAS II but a maximum of two electrons in RAS III. Because this is equivalent to a MR-CISD in which every possible active space configuration is used as a reference, SOCI represents an idealized limit of MR-CISD, which would be very difficult to employ for all but the smallest chemical systems. When the active space is small, SOCI describes the region near dissociation more accurately than that around equilibrium.

Assessment of the Accuracy of MRCI and CASPT2 This trend is seen for BH with SOCI (1:1) and HF with SOCI (val); the associated nonparallelity errors (1.5 and 3.2 kcal mol-1) are surprisingly large considering the very extensive description of electron correlation in SOCI. However, it must be kept in mind that the valence active space contains only one active unoccupied orbital for HF and the 1:1 active space contains no b1 or b2 orbitals for BH, which are important for dynamical correlation near equilibrium. When one uses the larger active space in each case, the SOCI parallels the FCI limit very well, as seen for BH with SOCI (val), HF with SOCI (1:1), and CH4 with SOCI (val ) 1:1). The nonparallelity errors for SOCI with the larger active space (0.29-0.04 kcal mol-1) are about an order of magnitude smaller than those for UCCSD(T) for these molecules.26 The significantly larger nonparallelity errors for the smaller active space show that even for very extensive MR-CISD computations, the active space must be chosen carefully if subchemical accuracy is to be achieved. Since its inception in 1990,48 many papers have been published using the CASPT2 variant of MRPT. Because CASPT2 is approximately size-consistent49 and will generally be more computationally efficient than MRCI, it may be attractive as an alternative approach for the determination of accurate potential energy surfaces. Previous studies48,50 have shown that the CASPT2 error mimics the trend of SOCIs maximum error near equilibrium and minimum error near dissociation. This trend is seen here in Figures 2 and 3. As one would expect, the error for CASPT2 decreases with the larger active space. In the case of HF, the larger active space also leads to a smaller NPE, but in BH, the NPE is hardly improved. It is interesting to note that the NPEs for CASPT2 are less sensitive to the active space than are the NPEs for SOCI. Except for HF with the valence active space, the CASPT2 NPEs are always larger than the SOCI NPEs, by a factor of around 2-10. This indicates that SOCI or similar large MR-CISD wave functions are preferable to CASPT2 for obtaining very accurate potential energy surfaces, such as might be required for highaccuracy vibrational level prediction. Nevertheless, the CASPT2 errors are modest, and they tend to compare well with CISD[TQ]. Compared to the single-reference results of our previous study, we find that CASPT2 does as well as UCCSD(T) for BH and much better for HF and CH4. We expect the improvement over UCCSD(T) to be larger for more challenging cases such as the homolytic dissociation of bonds between two nonhydrogen atoms. 4. Conclusions We have assessed various multireference methods for their performance in bond breaking reactions in BH, CH4, and HF. This represents the first detailed evaluation of multireference methods across the entire potential energy curve for single bondbreaking reactions, and it is made possible by our determination of computationally demanding full configuration interaction potential curves, which represent the exact solution for the given basis set. Although breaking bonds to hydrogen should represent one of the easiest types of bond-breaking reactions for theoretical methods, nevertheless our recent evaluation of single-reference methods indicated surprisingly large errors.26 For the present molecules, the multireference methods including dynamical correlation (CISD[TQ], SOCI, and CASPT2) all give more accurate results than the best standard single-reference method, UCCSD(T), even for these simple reactions. With current technology, multireference configuration interaction is the only method in common use for determining

J. Phys. Chem. A, Vol. 107, No. 29, 2003 5615 spectroscopic-quality potential energy surfaces. As the size of the system increases, the quality of the MRCI wave function will degrade, leading to a need for size-consistent (or at least approximately size-consistent) methods, including multireference perturbation theory or multireference coupled-cluster. The CASPT2 approach is a widely used multireference perturbation theory method which, according to the present study, appears to be somewhat less sensitive to the choice of active space than MRCI. However, the perturbative treatment of dynamical electron correlation does not seem as effective as extensive MRCI for the small systems considered here. Unfortunately, the prospects of improving the CASPT2 model by employing higher orders of perturbation theory does not seem promising, because Olsen and co-workers51 have shown that the multireference perturbation expansion does not converge. Nevertheless, if the proper active orbital space is chosen, CASPT2 is still preferred to spin-unrestricted single-reference methods for describing potential energy curves. Comparison of active space definitions, valence or one-toone, demonstrates that the best definition for a given molecule is that which produces the larger active space. The smaller active space led to relatively large errors in some cases. The ideal active space for multireference computations, which may be too large to employ in practice, would be one that is the union of the valence and one-to-one active spaces: that is, it should include all valence orbitals and ensure the presence of at least one active unoccupied orbital per active occupied orbital. Acknowledgment. C.D.S. is a Blanchard assistant professor at Georgia Tech, and he acknowledges a Camille and Henry Dreyfus New Faculty Award and an NSF CAREER Award (Grant No. CHE-0094088). The Center for Computational Molecular Science and Technology is funded through a Shared University Research (SUR) grant from IBM and by Georgia Tech. References and Notes (1) Hirao, K., Ed. Recent AdVances in Multireference Methods; Recent Advances in Computational Chemistry, Vol. 4; World Scientific: Singapore, 1999. (2) Schaefer, H. F. Ph.D. Thesis, Stanford University, Stanford, CA, 1969. (3) Huron, B.; Malrieu, J. P.; Rancurel, P. J. Chem. Phys. 1973, 58, 5745-5759. (4) Buenker, R. J.; Peyerimhoff, S. D. Theor. Chim. Acta 1974, 35, 33-58. (5) Andersson, K.; Roos, B. O. In Modern Electronic Structure Theory; Yarkony, D. R., Ed.; Advanced Series in Physical Chemistry, Vol. 2; World Scientific: Singapore, 1995; pp 55-109. (6) Hoffmann, M. R. In Modern Electronic Structure Theory; Yarkony, D. R., Ed.; Advanced Series in Physical Chemistry, Vol. 2; World Scientific: Singapore, 1995; pp 1166-1190. (7) Werner, H.-J. Mol. Phys. 1996, 89, 645-661. (8) Szalay, P. G. In Recent AdVances in Coupled-Cluster Methods; Bartlett, R. J., Ed.; Recent Advances in Computational Chemistry, Vol. 3; World Scientific: Singapore, 1997; pp 81-124. (9) Mahapatra, U. S.; Datta, B.; Mukherjee, D. In Recent AdVances in Coupled-Cluster Methods; Bartlett, R. J., Ed.; Recent Advances in Computational Chemistry, Vol. 3; World Scientific: Singapore, 1997; pp 155-181. (10) Adamowicz, L.; Malrieu, J.-P. In Recent AdVances in CoupledCluster Methods; Bartlett, R. J., Ed., Recent Advances in Computational Chemistry, Vol. 3; World Scientific: Singapore, 1997; pp 307-330. (11) Sherrill, C. D.; Schaefer, H. F. In AdVances in Quantum Chemistry; Lo¨wdin, P.-O., Ed.; Academic Press: New York, 1999; Vol. 34, pp 143269. (12) Celani, P.; Werner, H.-J. J. Chem. Phys. 2000, 112, 5546-5557. (13) Li, X.; Paldus, J. Mol. Phys. 2000, 98, 1185-1199. (14) Lischka, H.; Shepard, R.; Pitzer, R. M.; Shavitt, I.; Dallos, M.; Mu¨ller, T.; Szalay, P. G.; Seth, M.; Kedziora, G. S.; Yabushita, S.; Zhang, Z. Phys. Chem. Chem. Phys. 2001, 3, 664-673. (15) Andersson, K.; Roos, B. O. Int. J. Quantum Chem. 1993, 45, 591.

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